On vanishing of Kronecker coefficients (1507.02955v2)

Abstract: We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood-Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood-Richardson coefficients, unless P=NP. We also show that there exists a #P-formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that, despite the hardness of the positivity problem, there may well exist a positive combinatorial formula for the Kronecker coefficients. Finding such a formula is a major open problem in representation theory and algebraic combinatorics. Finally, we consider the existence of the partition triples $(\lambda, \mu, \pi)$ such that the Kronecker coefficient $k^\lambda_{\mu, \pi} = 0$ but the Kronecker coefficient $k^{l \lambda}{l \mu, l \pi} > 0$ for some integer $l>1$. Such "holes" are of great interest as they witness the failure of the saturation property for the Kronecker coefficients, which is still poorly understood. Using insight from computational complexity theory, we turn our hardness proof into a positive result: We show that not only do there exist many such triples, but they can also be found efficiently. Specifically, we show that, for any $0<\epsilon\leq1$, there exists $0<a<1$ such that, for all $m$, there exist $\Omega(2^{m^a})$ partition triples $(\lambda,\mu,\mu)$ in the Kronecker cone such that: (a) the Kronecker coefficient $k^\lambda{\mu,\mu}$ is zero, (b) the height of $\mu$ is $m$, (c) the height of $\lambda$ is $\le m^\epsilon$, and (d) $|\lambda|=|\mu| \le m^3$. The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.

• The paper reveals that deciding Kronecker coefficients' positivity is NP-hard, contrasting it with efficient tests for Littlewood-Richardson coefficients.
• The paper suggests a positive combinatorial formula by deriving a #P-formula for a specific class of Kronecker coefficients, offering new structural insights.
• The paper exposes saturation failures by constructing explicit examples where scaled coefficients become positive despite initial zeros, highlighting complex behavior.

Analysis of the Computational Complexity of Kronecker Coefficients

The paper by Ikenmeyer, Mulmuley, and Walter investigates the complexity of determining the positivity of Kronecker coefficients and provides several compelling insights into the problem. Prior beliefs suggested that this problem could be resolved within polynomial time, akin to the case with the Littlewood-Richardson coefficients. However, the authors establish that determining the positivity of Kronecker coefficients is NP-hard, highlighting a significant theoretical distinction between these two classes of coefficients in representation theory.

Key Results

1. Complexity Gap: The paper provides evidence that Kronecker coefficients are inherently more complex than Littlewood-Richardson coefficients. Littlewood-Richardson coefficients benefit from an efficient polynomial-time algorithm for deciding positivity, whereas Kronecker coefficients do not, under the assumption that P $\neq$ NP. This computational complexity gap introduces new challenges to prevalent theories in algebraic combinatorics and representation theory.
2. Potential for Positive Combinatorial Formulas: Although the decision problem is NP-hard, the authors suggest that there might exist a positive combinatorial formula for the Kronecker coefficients, perhaps somewhat analogous to the Littlewood-Richardson rule. A major open problem is whether this positive structure can be elucidated in a useful form, and the authors provide a $\#P$-formula for a specific class of Kronecker coefficients with this goal in mind.
3. Existence of "Holes" and Saturation Failures: The paper explores the phenomenon of "holes" — cases where $k^\lambda_{\mu, \pi} = 0$ while rescaled counterparts $k^{l\lambda}_{l\mu, l\pi} > 0$ for some integer $l > 1$. This indicates a failure of the saturation property for Kronecker coefficients and suggests a deeper structural intricacy not yet fully understood. The paper not only proves the existence of many such partition triples efficiently but also provides a construction schema that enables their explicit identification.

Implications and Future Directions

• Geometric Complexity Theory (GCT): The findings have significant implications for GCT and its approach to central computational complexity questions like the permanent versus determinant problem. The identification of NP-hardness in positivity questions suggests potential obstacles that GCT must overcome when aiming to prove superpolynomial lower bounds for computational problems.
• Representation Theory and Algebraic Combinatorics: This research reinforces the need for novel approaches in algebraic combinatorics to tackle the combinatorial complexity inherent in Kronecker coefficients. New combinatorial approaches and theoretical insights could significantly impact foundational understandings in representation theory.
• Computational Tools: Understanding the complexity landscape mapped out by these coefficients can guide the development of better computational tools and algorithms for manipulating and utilizing Kronecker coefficients in practical applications.
• Open Problems: The question of finding a tractable form of positive combinatorial formula for Kronecker coefficients persists. Further exploration may refine our understanding of the complexity classes these coefficients belong to, potentially leading to breakthroughs that can leverage existing polynomial-time frameworks.

Conclusion

This paper not only resolves significant questions about the complexity of Kronecker coefficients but also opens avenues for future research that challenge longstanding assumptions in theoretical computer science and representation theory. The crossover of computational complexity theory with classical mathematical disciplines shows promising directions for addressing these newly highlighted open problems. As the field progresses, these insights pave the way for deeper exploration into the algorithmic and combinatorial properties of representation-theoretic constructs.